From Quantum Chaos and Eigenstate Thermalization to Statistical Mechanics and Thermodynamics

TL;DR

This review surveys the eigenstate thermalization hypothesis (ETH) as a bridge between quantum chaos, random matrix theory, and statistical mechanics, detailing how isolated quantum systems thermalize and obey thermodynamic relations. It derives ETH and its thermodynamic consequences from chaotic eigenstates, discusses fluctuation theorems and drift-diffusion dynamics under driving, and presents universal energy distributions in driven contexts. It also covers integrable models via the generalized Gibbs ensemble (GGE) and generalized eigenstate thermalization, outlining when GGEs describe relaxation and how kinetic equations govern late-time approach to equilibrium. Together, the work provides a comprehensive framework linking microscopic quantum dynamics to macroscopic thermodynamics across chaotic and integrable regimes.

Abstract

This review gives a pedagogical introduction to the eigenstate thermalization hypothesis (ETH), its basis, and its implications to statistical mechanics and thermodynamics. In the first part, ETH is introduced as a natural extension of ideas from quantum chaos and random matrix theory (RMT). To this end, we present a brief overview of classical and quantum chaos, as well as RMT and some of its most important predictions. The latter include the statistics of energy levels, eigenstate components, and matrix elements of observables. Building on these, we introduce the ETH and show that it allows one to describe thermalization in isolated chaotic systems without invoking the notion of an external bath. We examine numerical evidence of eigenstate thermalization from studies of many-body lattice systems. We also introduce the concept of a quench as a means of taking isolated systems out of equilibrium, and discuss results of numerical experiments on quantum quenches. The second part of the review explores the implications of quantum chaos and ETH to thermodynamics. Basic thermodynamic relations are derived, including the second law of thermodynamics, the fundamental thermodynamic relation, fluctuation theorems, and the Einstein and Onsager relations. In particular, it is shown that quantum chaos allows one to prove these relations for individual Hamiltonian eigenstates and thus extend them to arbitrary stationary statistical ensembles. We then show how one can use these relations to obtain nontrivial universal energy distributions in continuously driven systems. At the end of the review, we briefly discuss the relaxation dynamics and description after relaxation of integrable quantum systems, for which ETH is violated. We introduce the concept of the generalized Gibbs ensemble, and discuss its connection with ideas of prethermalization in weakly interacting systems.

Figures (36)

• Figure 1: Examples of trajectories of a particle bouncing in a cavity: (a) non-chaotic circular and (b) chaotic Bunimovich stadium. The images were taken from scholarpedia stockmann_10.
• Figure 2: Nearest neighbor spacing distribution for the "Nuclear Data Ensemble" comprising 1726 spacings (histogram) versus normalized (to the mean) level spacing. The two lines represent predictions of the random matrix GOE ensemble and the Poisson distribution. Taken from Ref. bohigas_haq_83. See also Ref. guhr_muller_98.
• Figure 3: (Left panel) Distribution of 250,000 single-particle energy level spacings in a rectangular two-dimensional box with sides $a$ and $b$ such that $a/b=\sqrt[4]{5}$ and $ab=4\pi$. (Right panel) Distribution of 50,000 single-particle energy level spacings in a chaotic cavity consisting of two arcs and two line segments (see inset). The solid lines show the Poisson (left panel) and the GOE (right panel) distributions. From Ref. rudnik_08.
• Figure 4: The level spacing distribution of a hydrogen atom in a magnetic field. Different plots correspond to different mean dimensionless energies $\hat{E}$, measured in natural energy units proportional to $B^{2/3}$, where $B$ is the magnetic field. As the energy increases one observes a crossover between Poisson and Wigner-Dyson statistics. The numerical results are fitted to a Brody distribution (solid lines) brody_flores_81, and to a semi-classical formula due to Berry and Robnik (dashed lines) berry_robnik_84. From Ref. wintgen_friedrich_87.
• Figure 5: (a)--(g) Level spacing distribution of spinless fermions in a one-dimensional lattice with Hamiltonian \ref{['eq:fermionHam']}. They are the average over the level spacing distributions of all $k$-sectors (see text) with no additional symmetries (see Ref. santos_rigol_10a for details). Results are reported for $L=24$, $N=L/3$, $J=V=1$ (unit of energy), and $J'=V'$ (shown in the panels) vs the normalized level spacing $\omega$. The smooth continuous lines are the Poisson and Wigner-Dyson (GOE) distributions. (h) Position of the maximum of $P(\omega)$, denoted as $\omega_\text{max}$, vs $J'=V'$, for three lattice sizes. The horizontal dashed line is the GOE prediction. Adapted from Ref. santos_rigol_10a.